Review of Subdivision Schemes and their Applications

Background: Subdivision surfaces modeling method and related technology research gradually become a hot spot in the field of computer-aided design(CAD) and computer graphics (CG). In the early stage, research on subdivision curves and surfaces mainly focused on the relationship between the points, thereby failing to satisfy the requirements of all geometric modeling. Considering many geometric constraints is necessary to construct subdivision curves and surfaces for achieving high-quality geometric modeling. Objective: This paper aims to summarize various subdivision schemes of subdivision curves and surfaces, particularly in geometric constraints, such as points and normals. The findings help scholars to grasp the current research status of subdivision curves and surfaces better and to explore their applications in geometric modeling. Methods: This paper reviews the theory and applications of subdivision schemes from four aspects. We first discuss the background and key concept of subdivision schemes. We then summarize the classification of classical subdivision schemes. Next, we show the subdivision surfaces fitting and summarize new subdivision schemes under geometric constraints. Applications of subdivision surfaces are also discussed. Finally, this paper gives a brief summary and future application prospects. Results: Many research papers and patents of subdivision schemes are classified in this review paper. Remarkable developments and improvements have been achieved in analytical computations and practical applications. Conclusion: Our review shows that subdivision curves and surfaces are widely used in geometric modeling. However, some topics need to be further studied. New subdivision schemes need to be presented to meet the requirements of new practical applications.

The Family of Multiparameter Quaternary Subdivision Schemes

In the field of subdivision, the smoothness increases as the arity of schemes increases. The family of high arity schemes gives high smoothness comparative to low arity schemes. In this paper, we propose a simple and generalized formula for a family of multiparameter quaternary subdivision schemes. The conditions for convergence of subdivision schemes are also presented. Moreover, we derive subdivision schemes after substituting the different values of parameters. We also analyzed the important properties of the proposed family of subdivision schemes. After comparison with existing schemes, we analyze that the proposed family of subdivision schemes gives better smoothness and approximation compared with the existing subdivision schemes.

Lochkovian (Lower Devonian) conodonts from the Alengchu section, western Yunnan, China

The Lochkovian (Lower Devonian) conodont biostratigraphy in China is poorly known, and conodont-based subdivision schemes for the Lochkovian in peri-Gondwana (the Spanish Central Pyrenees, the Prague Synform, Sardinia, and the Carnic Alps) have not been tested in China. Therefore, we studied conodonts from the lower part (Bed 9 to Bed 13) of the Shanjiang Formation at the Alengchu section of Lijiang, western Yunnan to test the application of established subdivision schemes. The conodont fauna is assignable to 12 taxa belonging to eight genera (Ancyrodelloides, Flajsella, Lanea, Wurmiella, Zieglerodina, Caudicriodus, Pelekysgnathus, and Pseudooneotodus), and enables recognition of two chronostratigraphical intervals from the lower part of the Shanjiang Formation. The interval ranging from the uppermost part of Bed 9 to the upper part of Bed 10 belongs to the lower Lochkovian; whereas an interval covering the uppermost part of Bed 11 to the upper part of Bed 13 is correlated with the upper half of the middle Lochkovian. The Silurian-Devonian boundary is probably located within Bed 9, in the basal part of the Shanjiang Formation. However, the scarcity of specimens precludes definitive identification of bases of the lower, middle, and upper Lochkovian as well as other conodont zones recognized in peri-Gondwana.

A Family of Integer-Point Ternary Parametric Subdivision Schemes

New subdivision schemes are always required for the generation of smooth curves and surfaces. The purpose of this paper is to present a general formula for family of parametric ternary subdivision schemes based on the Laurent polynomial method. The different complexity subdivision schemes are obtained by substituting the different values of the parameter. The important properties of the proposed family of subdivision schemes are also presented. The continuity of the proposed family is C 2 m . Comparison shows that the proposed family of subdivision schemes has higher degree of polynomial generation, degree of polynomial reproduction, and continuity compared with the exiting subdivision schemes. Maple software is used for mathematical calculations and plotting of graphs.

Hermite B-Splines: n-Refinability and Mask Factorization

This paper deals with polynomial Hermite splines. In the first part, we provide a simple and fast procedure to compute the refinement mask of the Hermite B-splines of any order and in the case of a general scaling factor. Our procedure is solely derived from the polynomial reproduction properties satisfied by Hermite splines and it does not require the explicit construction or evaluation of the basis functions. The second part of the paper discusses the factorization properties of the Hermite B-spline masks in terms of the augmented Taylor operator, which is shown to be the minimal annihilator for the space of discrete monomial Hermite sequences of a fixed degree. All our results can be of use, in particular, in the context of Hermite subdivision schemes and multi-wavelets.

Clothoid fitting and geometric Hermite subdivision

We consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present numerical results produced by the proposed schemes and discuss their features.